Question

the palace hotel believes its customers may be waiting too long for room service. the hotel operations manager knows that the time for room service orders is normally distributed, and he sampled 10 room service orders during a 3-day period and timed each (in minutes), as follows: the operations manager believes that only 10% of the room service orders should take longer than 25 minutes if the hotel has good customer service. does the hotel room service meet this goal?

Answer 1

Based on the given sample data, we can conclude that the hotel room service does not meet the goal of having only 10% of orders taking longer than 25 minutes.

How to solve the problem

To determine whether the hotel room service meets the goal of having only 10% of orders taking longer than 25 minutes, perform a hypothesis test using the given sample data.

Given:

Sample size (n) = 10

Sample data: 23, 23, 15, 12, 26, 16, 18, 19, 30, 25

Let's set up the hypotheses:

Null Hypothesis (H0): The proportion of room service orders taking longer than 25 minutes is 10% or less (p ≤ 0.10).

Alternative Hypothesis (H1): The proportion of room service orders taking longer than 25 minutes is greater than 10% (p > 0.10).

Next, calculate the sample proportion (
`\hat{p}`) of room service orders taking longer than 25 minutes.

Counting the number of orders that took longer than 25 minutes from the given sample data, we have:

2 orders (23, 26, 30, and 25)

Therefore, the sample proportion is:

`\hat{p}` = 2/10 = 0.2

Use the normal approximation to perform the hypothesis test. Under the null hypothesis, the sampling distribution of the proportion can be approximated by a normal distribution.

To calculate the test statistic (z-score), use the formula:

z = (
`\hat{p}`t - p) /
`\sqrt`((p * (1 - p)) / n)

Substituting the values:

z = (0.2 - 0.10) /
`\sqrt`((0.10 * (1 - 0.10)) / 10)

Calculating this expression:

z ≈ 1.414

Next, find the critical z-value corresponding to a one-tailed test with a significance level of 0.10 (10%). This critical value represents the cutoff point beyond which we reject the null hypothesis.

Looking up the critical z-value from the standard normal distribution table or using statistical software, we find that the critical value for a one-tailed test with a significance level of 0.10 is approximately 1.282.

Since the calculated z-value (z ≈ 1.414) is greater than the critical z-value (1.282), we reject the null hypothesis.

Therefore, based on the given sample data, we can conclude that the hotel room service does not meet the goal of having only 10% of orders taking longer than 25 minutes.

The Palace Hotel believes its customers may be waiting too long for room service. The hotel operations manager knows that the time for room service orders is normally distributed, and he sampled 10 room service orders during a 3-day period and timed each (in minutes), as follows:

23 23

15 12

26 16

18 19

30 25

The operations manager believes that only 10% of the room service orders should take longer than 25 minutes if the hotel has good customer service. Does the hotel room service meet this goal?